Takagi-Sugeno Functional Fuzzy System for Function-on-Function Regression

Function-on-function (FOF) regression is a typical regression problem with the input and output being infinite dimensional functions. This research topic raises increasing attention in the statistical community. However, it is seldom investigated in the fuzzy systems community. In this paper, we propose a new fuzzy system model and its identification approach, the Takagi-Sugeno functional fuzzy system (T-S-FFS), for the FOF regression problem. Taking advantage of conventional fuzzy systems, T-S-FFS performs as a general nonlinear model constructed by a set of new T-S functional “If-then” fuzzy rules, which are developed based on the functional linear model having both the intercept and slope being functions. Inheriting the accuracy of T-S rules, such a rule base enables the model to be delicate and accurate; Furthermore, the slope function is easy to visualise, which helps in explaining the rules. In addition, to identify the parameters in T-S- FFS, we proposed a localized identification approach by solving ridge regression problems with L2 penalties. The proposed T-S-FFS model can achieve preferable predicting results compared with state-of-the-art models on benchmark examples.